Introduction Logic Inference. Discrete Mathematics Andrei Bulatov

Transcription

1 Introduction Logic Inference Discrete Mathematics Andrei Bulatov

2 Discrete Mathematics - Logic Inference 6-2 Previous Lecture Laws of logic Expressions for implication, biconditional, exclusive or Valid and invalid arguments Logic inference Rules of inference Modus Ponens Rule of Syllogism Modus Tollens Rule of Disjunctive Syllogism Rule for Proof by Cases Rule of Contradiction Rule of Simplification Rule of Amplification

3 Discrete Mathematics Logic Inference 6-3 Logic Inference The goal of an argument is to infer the required conclusion from given premises Formally, an argument is a sequence of statements, each of which is either a premise, or obtained from preceding statements by means of a rule of inference

4 Discrete Mathematics Logic Inference 6-4 Example Premises: ``It is not sunny this afternoon and it is colder than yesterday. We will go swimming only if it is sunny. If we do not go swimming, then we will take a canoe trip. If we take a canoe trip, then we will be home by sunset. Conclusion: ``We will be home by sunset. Notation: q - it is colder than yesterday r - we will go swimming p - it is sunny this afternoon s - we will take a canoe trip t - we will be home by sunset Premises: p q, r p, r s, and s t Conclusion: t

5 Discrete Mathematics Logic Inference 6-5 Example (cntd) We have p q, r p, r s, and s t Step Reason 1. p q premise 2. p simplification 3. r p premise 4. r modus tollens 5. r s premise 6. s modus ponens 7. s t premise 8. t modus ponens

6 Discrete Mathematics - Logic Inference 6-6 Logic Puzzles A prisoner must choose between two rooms each of which contains either a lady, or a tiger. If he chooses a room with a lady, he marries her, if he chooses a room with a tiger, he gets eaten by the tiger. The rooms have signs on them: I at least one of these rooms contains a lady II a tiger is in the other room It is known that either both signs are true or both are false

7 Discrete Mathematics - Logic Inference 6-7 Logic Puzzles (cntd) Notation: p - the first room contains a lady q - the second room contains a lady Premises: (p q) p, p (p q)

8 Discrete Mathematics - Logic Inference 6-8 Logic Puzzles (cntd) Argument Step Reason (p q) p, p (p q) 1. p (p q) premise 2. p p q expression for implication 3. p q idempotent law 4. (p q) p premise 5. p modus ponens 6. q rule of disjunctive syllogism to 3 and 5

9 Discrete Mathematics - Logic Inference 6-9 Conjunctive Normal Form A literal is a primitive statement (propositional variable) or its negation p, p, q, q A clause is a disjunction of one or more literals p q, p q r, q, s s r q A statement is said to be a Conjunctive Normal Form (CNF) if it is a conjunction of clauses p (p q) ( r p) p q ( r p) ( r q) (p q s r) ( r p) r

10 Discrete Mathematics - Logic Inference 6-10 CNF Theorem Theorem Every statement is logically equivalent to a certain CNF. Proof (sketch) Let Φ be a (compound) statement. Step 1. Express all logic connectives in Φ through negation, conjunction, and disjunction. Let Ψ be the obtained statement. Step 2. Using DeMorgan s laws move all the negations in Ψ to individual primitive statements. Let Θ denote the updated statement Step 3. Using distributive laws transform Θ into a CNF.

11 Discrete Mathematics - Logic Inference 6-11 Example Find a CNF logically equivalent to (p q) r Step 1. Step 2. ( p q) r (p q) r Step 3. (p r) ( q r)

12 Discrete Mathematics - Logic Inference 6-12 Rule of Resolution p q p r q r is called resolvent q r The corresponding tautology ((p q) ( p r)) (q r) ``Jasmine is skiing or it is not snowing. It is snowing or Bart is playing hockey. p - `it is snowing q - `Jasmine is skiing r - `Bart is playing hockey ``Therefore, Jasmine is skiing or Bart is playing hockey

13 Discrete Mathematics - Logic Inference 6-13 Computerized Logic Inference Convert the premises into CNF Convert the negation of the conclusion into CNF Consider the collection consisting of all the clauses that occur in the obtained CNFs Use the rule of resolution to obtain the empty clause ( ). If it is possible, then the argument is valid. Otherwise, it is not. Why empty clause? The only way to produce the empty clause is to apply the resolution rule to a pair of clauses of the form p and p. Therefore, the collection of clauses is contradictory. In other words, for any choice of truth values for the primitive statements, if premises are true, the conclusion cannot be false.

14 Discrete Mathematics - Logic Inference 6-14 Example A lady and a tiger Premises: p (p q), (p q) p Negation of the conclusion: q Clauses: p, p q, q Argument: p q premise resolvent q premise p resolvent p premise

15 Discrete Mathematics - Logic Inference 6-15 Homework Exercises from the Book: No. 5, 9a (page 84-85) - Prove that resolution is a valid rule of inference - Same arrangements as in the `A lady or a tiger problem. This time if a lady is in Room I, then the sign on it is true, but if a tiger is in it, then the sign is false. If a lady is in Room II, then the sign on it is false, and if a tiger is in it, then the sign is true. Signs are I both rooms contain ladies II both rooms contain ladies